几何增长系统中的幂律分布：COVID-19大流行的统计
The power-law distribution in the geometrically growing system: Statistic of the COVID-19 pandemic
AI总结：
本文提出几何增长系统产生幂律分布的一般机制，以log-CS$χ_1$分布近似并推导Zipf指数公式，用于解释COVID-19大流行统计。
摘要：
幂律分布无处不在，其形成机制似乎多种多样。我们发现了这种分布的一种一般机制。几何增长系统的分布可以用自由度为1的对数完全平方卡方分布（log-CS$χ_1$）来近似，该分布渐近地趋于幂律分布；或者在上限处用对数正态分布近似，后者具有无限的渐近斜率。对于log-CS$χ_1$，幂律的渐近指数或双对数图中的斜率似乎只与系统参数的方差及其相互相关性有关，而与系统的初始分布或任何参数均值无关。当系统必须具有奇异初始分布时，我们可以将log-CS$χ_1$作为唯一近似。该机制展现出可广泛应用于实践的全面性。我们推导了Zipf指数的一个简单公式，该公式可能要求指数接近-1而非恰好为-1。我们表明，这一方法可以解释COVID-19大流行的统计数据。
英文摘要：
The power-law distribution is ubiquitous and its mechanism seems to be various. We find a general mechanism for the distribution. The distribution of a geometrically growing system can be approximated by a log - completely squared chi distribution with 1 degree of freedom (log-CS$χ_1$), which reaches asymptotically a power-law distribution, or by a log-normal distribution, which has an infinite asymptotic slope, at the upper limit. For the log-CS$χ_1$, the asymptotic exponent of the power-law or the slope in a log-log diagram seems to be related only to the variances of the system parameters and their mutual correlation but independent of an initial distribution of the system or any mean value of parameters. We can take the log-CS$χ_1$ as a unique approximation when the system should have a singular initial distribution. The mechanism shows a comprehensiveness to be applicable to wide practice. We derive a simple formula for the Zipf's exponent, which will probably demand that the exponent should be near -1 rather than exactly -1. We show that this approach can explain statistics of the COVID-19 pandemic.
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